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We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geqC\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ whereeach $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ foreach $i, j\in [n]$, then asymptotically almost surely there is an order-$n$Latin square in which the entry in the $i$th row and $j$th column lies in$L_{i,j}$. The problem of determining the threshold probability for theexistence of an order-$n$ Latin square was raised independently by Johansson,by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result providesan upper bound which is tight up to a factor of $\log n$ and strengthens thebound recently obtained by Sah, Sawhney, and Simkin. We also prove analogousresults for Steiner triple systems and $1$-factorizations of complete graphs,and moreover, we show that each of these thresholds is at most the thresholdfor the existence of a $1$-factorization of a nearly complete regular bipartitegraph.

Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor